Dissecting Anomalies. Eugene F. Fama and Kenneth R. French. Abstract

Transcription

1 First draft: February 2006 This draft: June 2006 Please do not quote or circulate Dissecting Anomalies Eugene F. Fama and Kenneth R. French Abstract Previous work finds that net stock issues, accruals, momentum, profitability, and asset growth are associated with average returns that are unexplained by standard asset pricing models. We find that the anomalous average returns associated with net stock issues, accruals, and momentum are pervasive; they are strong in all size groups (tiny, small, and big) both in sorts and in cross-section regressions. In contrast, there is no asset growth anomaly in the average returns on the big stocks that account for more than 90% of total market cap. And profitability is consistently associated with anomalous returns in regressions but not in sorts. Graduate School of Business, University of Chicago (Fama) and Amos Tuck School of Business, Dartmouth College (French). We received helpful comments from seminar participants at the Swedish School of Economics.

2 There are patterns in average stock returns that are considered anomalies because they are not explained by the Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965). For example, Banz (1981) finds that stocks with low market capitalization (small stocks) have higher average returns than big (high market cap) stocks. There is much evidence that stocks with high ratios of book value to the market value of equity (value stocks) have higher average returns than low book-to-market (growth) stocks (Rosenberg, Reid, and Lanstein 1985, Chan, Hamao, and Lakonishok 1991, Fama and French 1992). Haugen and Baker (1996) and Cohen, Gompers, and Vuolteenaho (2002) find that more profitable firms have higher average stock returns. Fairfield, Whisenant, and Yohn (2003), and Titman, Wei, and Xie (2004) show that firms that invest more have lower stock returns. A literature initiated by Sloan (1996) finds that higher accruals predict lower stock returns. Pulling together earlier evidence that returns after stock repurchases are high (Ikenberry, Lakonishok, and Vermaelen 1995) and returns after stock issues are low (Loughran and Ritter 1995), Daniel and Titman (2005) show that there is a negative relation between net stock issues and average returns. The premier anomaly is momentum (Jegadeesh and Titman 1993): stocks with low returns over the last year tend to have low returns for the next several months and those with high past returns tend to have high future returns. This return momentum is left unexplained by the three-factor model of Fama and French (1993) as well as by the CAPM. We revisit the size, value, profitability, growth, accruals, net stock issues, and momentum anomalies. Each presents a path traveled by earlier work, but there are gains in studying them together to see which have information about average returns missed by the others. There are also methodology issues. For example, the anomalies literature tends to focus on extremes. A common approach is to form equal-weight (EW) decile portfolios by sorting all stocks on the variable of interest and then to focus on the hedge portfolio return obtained from long/short positions in the extreme deciles. A problem in this approach is that the returns on EW hedge portfolios that use all stocks are typically dominated by tiny stocks (which we define as stocks with market cap below the 20 th NYSE percentile).

3 Tiny stocks are influential in EW hedge returns for two reasons. First, though tiny stocks are on average only about 3% of the market cap of the NYSE-Amex-Nasdaq universe, they account for about 60% of the total number of stocks. Second, the cross-section dispersion of anomaly variables is largest among tiny stocks, so tiny stocks typically account for more than 60% of the stocks in extreme sort portfolios. The cross-section regression approach of Fama and MacBeth (FM 1973), which is also commonly used to predict returns with anomaly variables, is subject to the same problem. FM regressions give more weight to tiny stocks because they tend to have more extreme values of the explanatory variables and more extreme returns. Thus, FM regressions that combine tiny, small, and big stocks are likely to be dominated by the tiny stocks that are plentiful in our sample and in most previous work. The pervasiveness of anomaly returns across size groups is an important issue. From a practical perspective, the extreme returns on EW hedge portfolios that are common in the anomalies literature are probably not realizable because of the high costs of trading tiny stocks in high turnover EW portfolios. From a research perspective, it is also important to know whether anomalous patterns in returns are market wide phenomena or limited to illiquid stocks that represent only a small portion of market wealth. Focusing on extremes, whether explicitly in sorts or implicitly in cross-section regressions, may also miss important information in the returns associated with less extreme values of anomaly variables. For example, if an anomaly variable proxies for sensitivity to a risk factor in returns, we expect to see average returns that vary relatively smoothly across values of the variable. The same is true if an anomaly variable is a proxy for a behavioral bias that varies continuously with values of the variable. We examine anomaly returns in detail, using separate sorts and FM regressions for tiny, small, and big stocks (where the breakpoints are the 20 th and 50 th percentiles of market cap for NYSE stocks). The sorts examine the full spectrum of returns for each anomaly variable and for each size group. The FM regressions tell us which anomalies have marginal information about average returns not captured by 2

4 the rest. And diagnostic tests on the residuals allow us to judge whether the smooth relations between anomaly variables and returns implied by the regression slopes actually show up in average returns. We proceed as follows. Section I presents summary statistics for returns and the anomaly variables we use to predict returns. Section II examines the average returns produced by sorts of stocks on the anomaly variables. We find that net stock issues, accruals, and momentum produce strong abnormal EW and VW (value-weight) returns for tiny, small, and big stocks. There is, however, no asset growth anomaly in the average returns on the big stocks that account for more than 90% of total market cap, and at least in the sorts, profitability predicts returns only for small (not tiny or big) stocks. A general result from the sorts is that tiny stocks are influential in average EW returns for all stocks, and as usual, average VW returns for all stocks are dominated by big stocks. These results underscore our claim that for a full picture of anomaly returns, one must examine average returns for different size groups. Section III presents the cross-section regressions to identify which variables have information about average returns not captured by the rest. The three clear winners, in terms of pervasive effects across size groups, are again net stock issues, accruals, and momentum. But unlike the sorts, the regressions also suggest that there is a pervasive positive relation between profitability and average returns. Finally, analysis of the regression residuals produces little evidence of functional form problems. Thus, average returns seem to vary smoothly across values of the anomaly variables. Section IV summarizes the results and interprets them from the perspective of the standard valuation equation used, for example, in Fama and French (2005b). The valuation equation says that controlling for B/M, higher expected net cashflows (earnings minus investment, per dollar of book value) imply higher expected stock returns. We argue that all the anomaly variables are proxies for expected cashflows. We also argue that the observed relations between average returns and the anomaly variables (positive for momentum and profitability, negative for net stock issues, accruals, and asset growth) are in line with the valuation equation. 3

5 I. Summary Statistics At the end of June each year from 1963 to 2005 we allocate NYSE, Amex, and (after 1972) Nasdaq stocks to three size groups, tiny, small, and big, where the breakpoints are the 20 th and 50 th percentiles of end-of-june market cap for NYSE stocks. Tiny stocks correspond to what investment managers call micro-caps, which can be costly to trade. The small and big stock groups are also roughly in line with industry definitions. Table 1 shows averages and standard deviations of returns for the value-weight (VW) and equalweight (EW) tiny, small, and big portfolios for July 1963 to December 2005, along with time-series averages of the number of stocks and the percent of aggregate market cap in each portfolio. Results are also shown for the market portfolio of all sample stocks and for a sample that includes all but tiny stocks. On average, tiny stocks are 60% of all sample stocks, but they account for only about 3% of the market cap of stocks in the sample. Because tiny stocks are so plentiful, they are influential in the EW market return. With their high average EW return (1.56% per month, versus 1.07% for big stocks), tiny stocks pull the average EW market return up to 1.36% per month. The EW tiny portfolio also has by far the highest return volatility, and it dominates the volatility of the EW market return. In contrast, big stocks are more than 90% of total market cap, and they dominate VW market returns. The average monthly return and the standard deviation of the monthly return on the VW market (0.94% and 4.44%) are close to those of the VW big stock portfolio (0.92% and 4.36%). Table 1 also shows time-series averages of the standard deviations of the annual cross sections of returns and the anomaly variables we use to predict returns. (Detailed descriptions of the variables and our sample of firms are in the Appendix.) For returns and all anomaly variables, cross-section dispersion is largest for tiny stocks and declines from tiny to small to big stocks. And for returns and all anomaly variables, the cross-section standard deviations for tiny stocks dominate those for all stocks. This result is important because it implies that tiny firms have even more influence in standard marketwide anomaly 4

6 tests (EW average hedge returns from the extremes of sorts of all stocks and FM regressions estimated on all stocks) than their large numbers would imply. II. Sorts Table 2 shows average monthly VW and EW returns for July 1963 through December 2005 for sorts of tiny, small, and big stocks on each anomaly variable. For all but momentum, the sorts are done once a year at the end of June, and monthly returns are calculated from July through June of the following year. The monthly return on a stock is measured net of the return on a matching portfolio formed on size and book-to-market equity (B/M). The matching portfolios are the updated 25 VW size-b/m portfolios of Fama and French (1993), formed at the end of June each year, based on independent sorts of firms into market cap and B/M quintiles, using NYSE breakpoints for the quintiles. We refer to the adjusted average returns from the sorts as abnormal returns. They show the portion of anomaly average returns left unexplained by market cap and book-to-market equity. Skipping the details, we can report that these portfolio-adjusted average returns are similar to the intercepts from the three-factor regression model of Fama and French (1993) estimated on the portfolio returns from the anomaly sorts. Thus, Table 2 in effect shows average returns that cannot be explained by the three-factor model. Our portfolio-adjusted average returns are also similar to those obtained from anomaly sorts of the residuals from cross-section regressions of stock returns on (the logs of) market cap and B/M. We make two choices when setting sort breakpoints. First, to have meaningful comparisons of returns across size groups, the sorts for a variable use the same breakpoints for all size groups. The breakpoints are those for all but tiny stocks. The anomaly variables show more dispersion for tiny stocks, and if we include tiny stocks when setting sort breakpoints, we often have few small or big stocks in the extreme portfolios. Second, net stock issues, profitability, asset growth, and accruals take positive and negative values, and it is interesting to examine positives and negatives separately. But positives are more frequent than negatives. To produce cells that are more comparable in terms of number of stocks, 5

7 negative values of these variables are allocated to two groups, using the median of negative values of the variable for all but tiny stocks. Positive values are allocated to five groups, using the quintile breakpoints for positive values of the variable for all but tiny stocks. Many firms have no net stock issues, especially during the early years of the sample, so the sorts for net issues have an additional cell, for zeros. Except for size and momentum, the variables we use to forecast returns are measured with long lags relative to the returns. For portfolios formed in June of year t, variables from Compustat (book equity, B, and earnings, Y) are for the fiscal year ending in calendar year t-1, and Compustat variables that involve changes (asset growth, da/a; accruals, Ac/B; and net stock issues, NS) are changes from the fiscal year ending in calendar year t-2 to the fiscal year ending in calendar year t-1. Since the portfolios are formed once a year, the Compustat sort variables are from six to 30 months old when the returns they are used to predict are measured. This suggests that the anomaly returns we observe are persistent, either risk-related characteristics of expected returns or the result of behavioral biases that persist for rather long periods after the variable that signals the bias is observed. To separate out the effects of share issues and repurchases, the two accounting variables (accruals and asset growth) that are year-to-year changes are measured on a per share basis. The relation between average returns and share issues and repurchases is then captured by the net share issues variable, which is the change in the natural log of (split-adjusted) shares outstanding from the fiscal year ending in calendar year t-2 to the fiscal year ending in calendar year t-1. Size (market cap) and momentum, which use CRSP data, are measured in a more timely fashion than other variables. Size is measured once a year when portfolios are formed in June. Since momentum returns are short-term (Jegadeesh and Titman 1993), we measure momentum monthly. The momentum variable to predict returns for month j is the 11-month return for j-12 to j-2. We skip the return for the month before the return to be explained because of Jegadeesh s (1990) evidence of negative correlation of month-to-month returns the likely result of market microstructure effects (Jegadeesh and Titman 1995). 6

8 A. Overview We begin by discussing general results in the average returns from the sorts in Table 2. In Table 1, big stocks dominate VW market returns and tiny stocks dominate EW returns. These conclusions carry over when portfolios are formed by sorting stocks on the anomaly variables, and when returns on individual stocks are measured net of the returns on portfolios matched on size and B/M. With one exception, VW abnormal returns from the sorts of all stocks are close to VW abnormal returns from the sorts of big stocks. The exception is profitability, Y/B. Few big firms are highly unprofitable, so the VW abnormal return for the left cell of the profitability sorts of all stocks is not close to the corresponding VW abnormal return for unprofitable big firms. But for quintiles of positive profitability, VW abnormal returns in the sorts of all stocks are close to those for big stocks. The anomalies literature emphasizes hedge portfolio returns from long/short positions in the extreme portfolios from sorts of all stocks. The focus is typically on average EW hedge returns, with an aside for VW returns. Table 2 says that the EW hedge returns observed in such studies are heavily influenced by stocks that are tiny (not just small). The average EW hedge returns from sorts of all stocks are consistently closer to those for tiny stocks than to those for small or big stocks. It is by now clear why tiny stocks are influential in EW hedge returns for all stocks. On average, about 60% of the sample firms are tiny, so even if we formed sort portfolios randomly, about 60% of the stocks in each portfolio would be tiny. But the extreme portfolios from sorts of all stocks on the anomaly variables are not random. The average cross-section standard deviations of the anomaly variables (Table 1) are highest for tiny stocks, so tiny stocks are more likely than small or big stocks to end up in the extreme portfolios obtained from sorts of all stocks. As a result, tiny stocks are likely to dominate EW hedge returns from sorts of all stocks. Table 2 also says that EW abnormal returns for the sort portfolios of tiny stocks are typically much higher than VW abnormal returns. This is due to a strong size effect among tiny stocks (documented later): within the tiny group, tinier stocks have higher returns. In contrast, EW and VW 7

9 abnormal returns are more similar in the sorts of small stocks and in the sorts of big stocks. This reflects the fact (discussed later) that within the small and big groups the size effect is weak. B. Hedge Portfolio Returns Which anomalies produce strong average hedge returns for all three size groups? The clear winners in Table 2 are net stock issues, accruals, and momentum. The sorts on net issues produce negative abnormal hedge returns (the result of positive abnormal returns for large repurchases and negative returns for extreme issues) that are economically large (-0.54% to -0.71% per month) and more than 4.25 standard errors from zero for all three size groups and for EW and VW abnormal returns. Though smaller than net issue hedge returns, the negative average EW and VW hedge returns in the sorts on accruals (generated by positive abnormal returns for large negative accruals and negative returns for extreme positive accruals) are also economically large (-0.30% to -0.56% per month) and at least 3.30 standard errors from zero for tiny, small, and big stocks. Finally, momentum sorts produce large average VW and EW hedge returns for all size groups, and momentum hedge returns are typically larger than those for net issues and accruals. For example, the average monthly VW hedge returns for momentum are 1.37%, 1.16%, and 0.66% (t = 6.31, 5.28, and 2.73) for tiny, small, and big stocks. Momentum returns for tiny stocks merit comment. For every momentum quintile, the EW abnormal return for tiny stocks is higher than the matching VW return. This is in part the result of a strong size effect among tiny stocks, documented later. But a size effect does not explain why the spread in VW momentum returns for tiny stocks is near twice as high (1.37% per month, t = 6.31) as for EW returns (0.73%, t = 3.39), largely due to a slightly positive EW abnormal return for extreme losers, which turns strongly negative in VW returns. It appears that there is less momentum among the tiniest tiny losers than in the bigger tiny stocks that dominate VW returns. This result, foreshadowed in Hong, Stein, and Lim (2000), creates an interesting complication for theories intended to explain momentum returns. The results so far say that if one bases inferences on the hedge returns associated with extreme values of anomaly variables, then net stock issues, accruals, and momentum pass the pervasiveness test; 8

10 they produce large average EW and VW hedge returns in all size groups. Anomalous returns are less pervasive for asset growth. Sorts on asset growth produce large negative average hedge returns (high abnormal returns for large declines in assets and low abnormal returns for large increases) for tiny and small stocks, but not for the big stocks that account for more than 90% of total market cap. The average VW spreads for tiny and small stocks are -0.57% and -0.31% per month (t = and -2.43), but the average spread for big stocks is near zero (-0.02%, t = -0.10). The average EW spread for tiny stocks, -0.92% per month, is more than 50% larger than the VW spread, so it seems that the inverse relation between the asset growth anomaly and size holds even among tiny stocks. Profitability sorts produce the weakest average hedge portfolio returns. Only the small group produces abnormal hedge returns (EW and VW) more than two standard errors from zero. Thus, hedge returns do not produce much basis for the conclusion that, with controls for market cap and B/M, there is a positive relation between average returns and profitability. The evidence of a positive relation is, however, stronger in the cross-section regressions below. C. The Spectrum of Anomaly Returns Hedge portfolios focus on the extremes of the sorts on anomaly variables, so they do not give a full picture of the spectrum of abnormal returns. The issue is important for both risk-based and behavioral explanations of return anomalies; we expect to see average returns that vary smoothly across values of an anomaly variable if the variable is a good proxy for sensitivity to a risk factor in returns or if anomalous returns are due to behavioral biases that vary continuously with the variable. In short, for a full picture of the average returns associated with an anomaly variable, results for different size groups and the full spectrum of values of the variable are pertinent. The smoothness of average returns from a sort is difficult to judge without information about how the anomaly variable itself varies across the cells of the sort. Table 3 shows time-series averages of the annual averages and standard deviations of the anomaly variables within the cells of the Table 2 sorts. A clear result in Table 3 is that much of the action in anomaly variables is in the extremes. The jumps in the 9

11 average values of the variables from the extreme cells of the sorts to adjacent interior cells dwarf the changes across interior cells. The standard deviations of the anomaly variables within the extreme cells of the sorts are also several times those of interior cells. These results are not surprising. They just say that the distributions of anomaly variables are not uniform; they are thinner in the extremes, so the extreme cells of the sorts cover wider ranges of the variables. We shall see, however, that most (but not all) anomaly variables have interesting variation across interior cells of the sorts. And for some variables, firms in the extremes are quite unusual. We comment on these issues below, in the course of discussing the spectrum of anomaly returns. Which anomalies are present in all size groups and produce abnormal returns that vary smoothly from the low to the high ends of the sorts? Momentum satisfies both criteria. VW abnormal momentum returns are strongest for tiny stocks and weakest for big stocks, but they are impressive in all size groups, and they increase rather smoothly from strongly negative for extreme losers to strongly positive for extreme winners. EW momentum returns in all size groups also vary smoothly from losers to winners. For net stock issues, average EW and VW hedge returns from the extremes of the sorts are strong for all size groups, but abnormal returns do not vary much across most of the interior cells of the sorts. For judging abnormal returns across the spectrum of a variable, weighting stocks equally seems more relevant than value weighting. In all size groups, extreme negative net issues (percent repurchases above the median for all but tiny stocks) are associated with strong positive EW abnormal returns (0.47%, 0.26%, and 0.28% per month for tiny, small, and big stocks). EW abnormal returns are smaller (0.42%, 0.15%, and 0.17%), but still statistically reliable (t = 4.43, 1.94, and 2.88) for less extreme repurchases. Thus, positive abnormal returns after repurchases are pervasive. The problem is that for all size groups, the first three quintiles of positive stock issues also have positive EW abnormal returns, and they are near as large as those for small repurchases. EW abnormal returns decline for the fourth quintile of positive net issues, but only the highest quintile produces abnormal returns well below the returns on portfolios matched on size and B/M. 10

12 Our positive EW abnormal returns for repurchases are consistent with existing event study evidence, and our new result that abnormal returns are more extreme for larger repurchases fits nicely with earlier evidence. But our positive EW abnormal returns over most of the range of positive net issues seem to contradict previous studies. Except for Daniel and Titman (2005), however, previous papers on the returns after stock issues are event studies that focus on announcements of seasoned equity offerings (SEOs) or stock issues to complete mergers. These events probably fall into the fourth and especially the fifth quintile of positive net issues, which show large average values of net issues (Table 3), and they probably account for the strong negative abnormal returns observed in the fifth quintile (Table 2). Fama and French (2005a) find that though SEOs and stock-financed mergers are infrequent, net issues of stock are common. Other ways of issuing stock include executive options, grants, and other employee benefit plans, conversions of debt and preferred stock, warrants, rights issues, and direct purchase plans. The positive abnormal returns for the less extreme net stock issue quintiles are probably associated with these more common activities. Whatever the source, the novel finding in Table 2 is that the first three quintiles of positive net issues of stock are associated with positive abnormal returns, and consistent negative abnormal returns are limited to the extreme quintile of issues. Table 3 provides additional perspective on these results. Repurchases above the median average about 5% of stock outstanding, but repurchases below the median average only 0.5% of outstanding stock. The first three quintiles of positive net issues also involve small amounts of stock (on average about 0.1%, 0.6%, and 1.5% of stock outstanding). Thus, the positive abnormal returns for the first three quintiles are associated with rather minor issuing events. But net stock issues average a substantial 4.5% of stock outstanding in the fourth quintile of positive net issues. The fact that EW abnormal returns for this quintile are positive for tiny stocks and close to zero for small and big stocks is a problem for theoretical models that predict negative returns after stock issues. The strong negative abnormal returns for stock issues predicted by these stories are limited to the fifth quintile of positive net issues, where issues average an impressive 20% to 26% of shares outstanding. 11

13 Like net stock issues, the accruals sorts create large EW and VW abnormal hedge returns for all size groups. Extreme negative accruals are followed by strong positive abnormal returns, and extreme positive accruals are followed by strong negative abnormal returns. But less extreme accruals, positive or negative, tend to be followed by positive abnormal EW returns that do not decline much with the size of accruals. Except for tiny stocks, however, the EW abnormal returns associated with less extreme accruals are rather close to zero. Unlike net stock issues, however, there is lots of variation in accruals across the interior cells of the sorts. We measure accruals as the change in operating working capital per (split-adjusted) share outstanding divided by book equity per share. The average value of accruals for the fourth quintile of positive values is about 11% of book equity, which is close to the average profitability of small and big stocks (Table 1). The average values of accruals for the second and third quintiles of positive values (3.2% and 6.1% of book equity) are also nontrivial. Firms with extreme accruals are quite unusual. Accruals in the fifth quintile of positive values average more than 28% of book equity, and accruals below the median of negative values average % (big stocks) to % (small stocks) of book equity. It is interesting that these apparently unusual firms are the whole story for the accruals anomaly in returns. Sloan (1996) and most of the follow-up papers on accruals explain large hedge portfolio returns as the result of a behavioral bias. Investors do not understand that accruals mean revert faster than the cash component of earnings. As a result, investors overestimate the future earnings of firms with high current accruals and underestimate earnings for firms with negative accruals. Our results contradict this story, unless the behavioral biases it proposes are limited to the extremes of accruals, or unless the mean reversion of accruals is limited to the extremes. Not surprisingly, the details of the sorts on asset growth confirm the inference from hedge returns that this candidate anomaly does not produce pervasive abnormal returns. In the asset growth sorts, tiny and small stocks produce rather large average EW and VW hedge portfolio returns. And abnormal 12

14 returns for tiny stocks fall rather smoothly from extreme declines to extreme increases in assets. But this pattern is absent from the average returns on small and big stocks. Thus, only tiny stocks seem to produce a smooth negative relation between asset growth and abnormal returns. Finally, since profitability sorts produce weak results for hedge returns, it is not surprising that the details of the sorts also produce little evidence of pervasive abnormal returns. Only the sorts for small stocks produce average returns that increase smoothly from unprofitable to extremely profitable firms. There is no consistent pattern for big stocks, and for tiny stocks the ordering of average abnormal returns changes from decreasing for EW returns (the most unprofitable firms have the highest subsequent returns) to increasing for VW returns. Note, however, that if we restrict attention to firms with positive profitability, abnormal returns in all size groups do tend to increase with profitability. This result is important in evaluating the regression evidence that follows. III. Cross-Section Regressions Which anomalies are distinct and which have little marginal ability to predict returns? We use the cross-section regression approach of Fama and MacBeth (FM 1973) to answer this question. FM regressions face potential problems. Unlike sorts, regressions impose a functional form on the relation between anomaly variables and returns. This structure is what gives regressions the power to disentangle the return effects of multiple anomalies. The functional form may, however, be incorrect. To address this issue, we examine sorts of regression residuals on each explanatory variable. The residual sorts also allow us to examine whether returns vary smoothly with the anomaly variables, in the manner predicted by the regression slopes. Another problem is that, as in the marketwide sorts, tiny stocks are likely to dominate FM regressions estimated on all stocks. There are three reasons. First, on average about 60% of all sample stocks are tiny. Second, the cross-section standard deviations of the anomaly variables the independent variables in the FM regressions are highest for tiny stocks (Table 1). This means tiny stocks have an impact on regression slopes beyond that implied by their large numbers. Third, regression slopes 13

15 minimize residual variance, so extreme residuals have a bigger impact on regression estimates. The average of the cross-section standard deviations of monthly returns on tiny stocks (17.51%) is about 50% larger than that of small stocks (11.41%) and about twice that of big stocks (8.77%). For a more balanced picture, we fit FM regressions separately for tiny, small, and big stocks, as well as for all stocks and a sample that excludes tiny stocks. Estimating separate regressions for different size groups also allows simple difference-of-means tests of whether the expected values of regression slopes differ across groups. The structure of the regressions is similar to the structure of the sorts. Although we estimate the regressions monthly, we again update most of the explanatory variables once a year. Thus, we explain the cross section of monthly returns from July of year t to June of t+1 using anomaly variables observed in June of t or earlier. The exception to this rule is the momentum variable, which we update monthly. Our first tests, in Table 4, are baseline regressions that use market cap and B/M to forecast returns. Table 4 also examines the effect of augmenting market cap and B/M with one explanatory variable (net stock issues, accruals, or momentum), or a natural subset of variables (profitability and asset growth, used together in Fama and French 2005b). The results of Fama and French (2005b) also lead us to (i) estimate profitability slopes using only positive values of profitability, covering negative values with a dummy variable, and (ii) estimate separate slopes for positive and negative accruals. Finally, net stock issues are zero for many firms, especially during the early years of the sample, and all regressions that include net issues use a dummy variable to isolate zeros. The purpose of the Table 4 regressions is to examine which anomaly variables, used alone or in natural subsets, show power to describe the cross section of average stock returns. Our primary interest is an overall regression (Table 5) that includes all explanatory variables, and so allows inferences about each variable s marginal information about average returns. It turns out, however, that inferences are largely unchanged in going from the partial (Table 4) to the full (Table 5) regressions. Thus, we begin by discussing the Table 4 regressions in detail. 14

16 A. Size and Book-to-Market As a warm up, we examine the average slopes from the regressions that use (the natural logs of) market cap and B/M to predict returns. Beginning with Fama and French (1992), this is a common regression in the asset-pricing literature. Table 4 shows, however, that there is something to be learned, in particular about the size effect, from estimating the regressions separately for different size firms. Note first that, like previous work, the regressions that use all stocks produce strong average slopes, negative for market cap (-0.15, t = -2.85) and positive for B/M (0.33, t = 4.46). The novel evidence is that these results draw much of their power from tiny stocks. Most striking is the absence of a size effect in the regressions for small and big stocks and in the regression that uses all but tiny (that is, small and big) stocks. This result is not due to the power lost from splitting the sample into size groups. The average market cap slopes for small and big stocks, 0.04 and -0.05, are 5.33 and 4.40 standard errors below the average for tiny stocks, In short, tiny stocks are influential in the size effect observed in tests on all stocks. The relation between average returns and B/M is more robust. The average slopes for B/M are similar for tiny and small stocks, 0.34 (t = 4.47) and 0.32 (t = 3.47), but the average slope for big stocks (0.15, t = 1.58) is less than half as large. The difference-of-means tests show that the average B/M slope for big stocks is more than 2.4 standard errors below the slopes for tiny and small stocks. Fama and French (2005c) find, however, that the weaker relation between average returns and B/M for big stocks is special to the post-1962 period and to U.S. stocks. B. Initial Regressions for Subsets of Anomaly Variables Added individually to the regressions that use size and B/M to describe returns, net stock issues, momentum, and positive accruals show strong explanatory power in all size groups. The consistency of the slopes for non-zero net stock issues is impressive. The average slopes for non-zero issues are close to -2.0 for tiny, small, and big stocks, all are more than -4.7 standard errors from zero, and they differ from one another by at most 0.56 standard errors. Table 4 thus says that, controlling for size and B/M, larger 15

17 net issues of stock are associated with lower future returns, and the relation between issues and returns is much the same for tiny, small, and big stocks. The average slopes for the dummy variable for zero net stock issues range from (t = -4.40) for tiny stocks to (t = -1.81) for big stocks (Table 4). We find later, however, that the average slopes are closer to zero in multiple regressions that include all anomaly variables. The average slopes for positive accruals in Table 4 are negative, they are at least 1.82 standard errors below zero for all size groups, and they are within 0.56 standard errors of one another. Thus higher positive accruals are consistently associated with lower future average returns. The average slopes for negative accruals are mostly negative, but all are within one standard error of zero. This result seems puzzling given the evidence from the sorts (Table 2) that negative accruals are followed by rather strong positive average EW returns. Inspection of the sorts suggests an explanation: as the level of negative accruals approaches zero, the level of average returns does not drop off enough to generate a reliably negative average regression slope. If this is correct, the regressions may be improved by replacing negative accruals with a dummy variable. Like the sorts, the regressions say that the positive relation between average returns and momentum is strong for all size groups, but average momentum slopes differ across groups. In particular, the average slope for tiny stocks (0.43, t = 3.00) is about half the size and more than 2.5 standard errors below the slopes for small and big stocks (0.84, t = 5.14, and 0.83, t = 4.24). These results suggest that if the momentum anomaly is due to a risk factor in returns, the relation between our momentum variable and sensitivity to the factor varies across size groups. The much smaller average momentum regression slope for tiny stocks in Table 4 seems at odds with the near identical spreads in average EW momentum returns for tiny and big stocks in the sorts of Table 2. Table 1 shows, however, that the cross-section standard deviation of the momentum variable is on average about 50% larger for tiny stocks than for big stocks. Table 3 confirms that average values of the momentum variable in the extreme cells of the sorts are more extreme for tiny stocks. Larger spreads 16

18 in the momentum variable for tiny stocks versus big stocks combine with similar spreads in EW average sort returns to produce lower average slopes for the momentum variable in the cross-section regressions for tiny stocks. Note, however, that all approaches (regressions and EW and VW sort returns) agree that there are strong relations between momentum and average returns in all size groups. The regressions in Table 4 confirm the inference from the sorts in Table 2 that the relation between average returns and the growth rate of assets is not pervasive. Among tiny and small stocks, faster asset growth is associated with lower future returns. The average slopes for asset growth in the regressions for tiny and small stocks are 7.14 and 3.40 standard errors below zero. But the average slope for big stocks is much smaller, less than one standard error from zero, and more than two standard errors below the average slopes for tiny and small stocks. Thus, like the sorts, the regressions for big stocks do not identify a reliable relation between average returns and asset growth. Finally, like the sorts, the regressions say that profitable small stocks produce a reliable positive relation between profitability and average returns; the average Pos Y/B slope for small stocks is 2.80 standard errors from zero. But the regressions also produce evidence of positive relations between profitability and future returns for profitable tiny and big stocks. The average Pos Y/B slope for tiny stocks, though about half the slope for small stocks, is 1.77 standard errors above zero, and the slope for big stocks is 1.68 standard errors from zero. Moreover, the differences between the average slopes for the size groups are all within 1.32 standard errors of zero. Given this result, it is reasonable to point to the strong average slope estimated for all stocks (1.01, t = 3.37) as reliable evidence of an overall positive relation between positive profitability and average returns. (The average slopes in Table 4 for the dummy variable for unprofitable firms are essentially zero.) Is the regression evidence for profitability in conflict with the sorts? The positive relation between profitability and average returns observed for all size groups in the regressions is estimated using firms with positive profitability. The absence of profitability effects for tiny and big stocks in the sorts comes from hedge returns from extremely profitable and extremely unprofitable firms. As noted earlier, 17

19 if we look only at profitable firms, the sorts also suggest positive relations between profitability and average returns in all size groups. Thus, the open puzzle in both the regressions and the sorts is the absence of systematic relations between negative profitability and average returns for tiny and big firms. C. Multiple Regressions that Include all Anomaly Variables Are the return anomalies in Tables 2 and 4 distinct or do different variables provide correlated information about future returns? The multiple regressions in Table 5 address this issue. The only variable that does not suffer when placed in competition with others is market cap. The market cap slopes for all size groups in Table 5 are more negative than in Table 4, and the market cap slope for big stocks is standard errors from zero in Table 5. But Table 5 confirms that the size effect is strongest, by far, among tiny stocks. For variables other than market cap, the average slopes in the full regressions of Table 5 are typically closer to zero than the slopes in Table 4. (The rare exceptions are all cases where the average slopes in Table 5 are trivially different from those in Table 4.) Despite the attenuation of the slopes in the full regressions, the inferences from Table 4 carry over to Table 5. Net stock issues, positive accruals, and momentum continue to show up strongly in the return regressions for all size groups. The relations between average returns and asset growth are again special to tiny and small stocks. And the regressions again suggest that in all size groups more profitable firms have higher average returns. One result in Table 4 is weaker in Table 5. Specifically, the evidence in Table 4 that firms that do not issue or retire stock (Zero NS) have lower average returns largely disappears in the full regressions of Table 5. The average slope for Zero NS for small stocks is essentially zero in Table 5, the slopes for tiny and big stocks fall by about half, and only the slope for tiny stocks is more than two standard errors from zero. Thus, firms with zero net stock issues apparently tend to have other characteristics that predict lower average stock returns. In general, however, the links between anomaly variables and future returns in the partial regressions of Table 4 survive when we combine all the anomaly variables in the regressions in Table 5. 18

20 Thus, the anomaly variables we examine are not redundant; those that have meaningful return information in the sorts and partial regressions continue to make distinct contributions in the full regressions. D. Regression Diagnostics Except for profitability, accruals, and dummy variables, the explanatory variables in the regressions are natural logs (market cap, B/M, momentum) or changes in logs (asset growth, net stock issues). And the regression explanatory variables are winsorized at the 0.5 and the 99.5 percentiles. Are the resulting regressions well-specified? Or is there evidence that returns do not vary smoothly with our versions of the anomaly variables? To answer this question, we use univariate sorts on the anomaly variables to assign firms to groups and then examine each group s EW average residual from the full regressions in Table 5. This is analogous to the sorts that look at EW abnormal returns in Table 2. In fact, since we again use all but tiny stocks to determine the breakpoints for the residual tests, firms are assigned to the same groups in the residual sorts (Table 6) as they are in the return sorts (Table 2). Note first that the average residuals from the sorts are typically estimated precisely. For example, in the regression for tiny stocks, the average monthly residual for the highest quintile of accruals, -0.09% (about 1% per year), is modest in economic terms, but it is three standard errors below zero. In judging the results in Table 6, we balance economic against statistical significance. In general, the average residuals in Table 6 say that the full regressions in Table 5 absorb the abnormal returns observed in the sorts in Table 2. The average residuals nevertheless identify a minor problem. Three anomaly variables (net stock issues, accruals, and momentum) produce large spreads in abnormal returns in the sorts for all size groups in Table 2. The average residuals in the extremes of the sorts on these variables in Table 6 are typically close to zero, but they almost always have the same sign as the abnormal returns in Table 2. For example, when sorted on accruals, the average residuals in Table 6 are small in economic terms. But like the abnormal returns from the sorts in Table 2, extreme negative accruals are associated with positive average residuals in Table 6, and the average residuals for extreme positive accruals are 19

21 negative. Similarly, though the average residuals in the sorts on net stock issues are typically small in economic terms, the average residuals for firms that repurchase stock almost always reproduce the positive sign of the much larger abnormal returns for repurchases in Table 2. And the regression residuals for the fourth and fifth quintiles of positive net issues reproduce the negative sign of the much more extreme abnormal returns observed in the net issue sorts of Table 2. Momentum poses the biggest challenge to regression specification. In the sorts of residuals on momentum, the average residuals for the highest momentum cell are positive for all size groups and they are rather large for tiny (0.22% per month, t = 4.37) and small (0.11%, t = 2.91) stocks. Thus, the full regressions absorb most of the spread in average returns related to momentum (compare Tables 2 and 6), but some of the high returns of extreme positive momentum stocks are left in the regression residuals. We noted earlier that much of the action in anomaly average returns and in the anomaly variables themselves is in the extremes (see Tables 2 and 3). Thus, the average regression slopes in Table 5 put lots of weight on absorbing returns in the extremes, and Table 6 says the effort is largely successful. But Table 6 also says that the regressions move abnormal returns close to zero across the spectrum of every anomaly variable. For net stock issues and accruals, this result is a bit surprising. Specifically, in Table 2 EW abnormal returns (adjusted for market cap and B/M effects) for net stock issues and accruals only turn systematically negative for the fifth quintiles of the variables. It then might seem surprising that the negative slopes for positive values of these variables reduce abnormal returns across the board, not just in the extreme quintiles. Adding variables to regressions, however, affects all coefficients (including intercepts). We can infer from the average residuals in Table 6 that despite negative slopes for net issues and accruals, the full regressions actually predict higher average returns across most of the spectrum of positive net issues and accruals than controls for market cap and B/M alone. In sum, the full regressions in Table 5 produce average regression residuals in Table 6 that are almost always trivial in magnitude and closer to zero across the full range of each anomaly variable than the average abnormal returns in Table 2. This implies that average returns vary rather smoothly across 20

22 values of the anomaly variables, in the manner predicted by the regression slopes. The impression from Table 2 that for many anomalies the action in abnormal returns seems to be concentrated in the extremes then simply reflects the fact (Table 3) that much of the action in the variables themselves is in the extremes. Overall, it does not seem that much is to be gained by attempting to improve regression specification, except perhaps for momentum. IV. Conclusions and Interpretation Previous work finds that net stock issues, accruals, momentum, profitability, and asset growth are associated with average returns that are not completely explained by standard asset pricing models. Earlier tests, however, are sometimes dominated by tiny stocks. We explore the pervasiveness of return anomalies by examining tiny, small, and big stocks separately. In univariate sorts, net stock issues, accruals, and momentum are associated with abnormal returns that show up in all size groups (tiny, small, and big). In contrast, there is no evidence of an asset growth anomaly in the average returns on the big stocks that account for more than 90% of total market cap. Among profitable firms, higher profitability is associated with higher abnormal returns, but only small stocks produce evidence of an anomalous relation between profitability and average returns that extends to unprofitable firms. These inferences from the sorts carry over to full regressions that include all anomaly variables. The full regressions say that each of the anomaly variables we consider seems to have unique information about future returns. This conclusion does not mean we lack a unifying logic for the anomalies. In fact, the evidence from the sorts and the regressions is consistent with the standard valuation equation which says that controlling for B/M, higher expected net cashflows (earnings minus investment, per dollar of book value) imply higher expected stock returns whether the pricing of securities is rational or irrational. (See Fama and French 2005b for details.) All the anomaly variables are proxies for expected cashflows. For example, firms that repurchase stock tend to have higher net cashflows (high earnings relative to investment), and the reverse is true for 21